First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation ∂/∂t=± ∂ N /∂ x N
Abstract
Consider the high-order heat-type equation
∂u/∂t=±
∂
N
u/∂
x
N
for an integer
N>2
and introduce the related Markov pseudo-process
(X(t)
)
t≥0
. In this paper, we study several functionals related to
(X(t)
)
t≥0
: the maximum
M(t)
and minimum
m(t)
up to time
t
; the hitting times
τ
+
a
and
τ
−
a
of the half lines
(a,+∞)
and
(−∞,a)
respectively. We provide explicit expressions for the distributions of the vectors
(X(t),M(t))
and
(X(t),m(t))
, as well as those of the vectors
(
τ
+
a
,X(
τ
+
a
))
and
(
τ
−
a
,X(
τ
−
a
))
.